Properties

Label 8-189e4-1.1-c1e4-0-0
Degree $8$
Conductor $1275989841$
Sign $1$
Analytic cond. $5.18747$
Root an. cond. $1.22848$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  − 2·4-s − 8·7-s − 5·16-s + 10·25-s + 16·28-s − 4·37-s + 8·43-s + 34·49-s + 20·64-s − 40·67-s + 8·79-s − 20·100-s − 28·109-s + 40·112-s + 34·121-s + 127-s + 131-s + 137-s + 139-s + 8·148-s + 149-s + 151-s + 157-s + 163-s + 167-s + 28·169-s − 16·172-s + ⋯
L(s)  = 1  − 4-s − 3.02·7-s − 5/4·16-s + 2·25-s + 3.02·28-s − 0.657·37-s + 1.21·43-s + 34/7·49-s + 5/2·64-s − 4.88·67-s + 0.900·79-s − 2·100-s − 2.68·109-s + 3.77·112-s + 3.09·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.657·148-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + 2.15·169-s − 1.21·172-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{12} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{12} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(3^{12} \cdot 7^{4}\)
Sign: $1$
Analytic conductor: \(5.18747\)
Root analytic conductor: \(1.22848\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{189} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 3^{12} \cdot 7^{4} ,\ ( \ : 1/2, 1/2, 1/2, 1/2 ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(0.3053494770\)
\(L(\frac12)\) \(\approx\) \(0.3053494770\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$\Gal(F_p)$$F_p(T)$
bad3 \( 1 \)
7$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
good2$C_2^2$ \( ( 1 + T^{2} + p^{2} T^{4} )^{2} \)
5$C_2^2$ \( ( 1 - p T^{2} + p^{2} T^{4} )^{2} \)
11$C_2^2$ \( ( 1 - 17 T^{2} + p^{2} T^{4} )^{2} \)
13$C_2^2$ \( ( 1 - 14 T^{2} + p^{2} T^{4} )^{2} \)
17$C_2$ \( ( 1 + p T^{2} )^{4} \)
19$C_2$ \( ( 1 - 7 T + p T^{2} )^{2}( 1 + 7 T + p T^{2} )^{2} \)
23$C_2^2$ \( ( 1 - 41 T^{2} + p^{2} T^{4} )^{2} \)
29$C_2^2$ \( ( 1 - 38 T^{2} + p^{2} T^{4} )^{2} \)
31$C_2$ \( ( 1 - 11 T + p T^{2} )^{2}( 1 + 11 T + p T^{2} )^{2} \)
37$C_2$ \( ( 1 + T + p T^{2} )^{4} \)
41$C_2^2$ \( ( 1 + 67 T^{2} + p^{2} T^{4} )^{2} \)
43$C_2$ \( ( 1 - 2 T + p T^{2} )^{4} \)
47$C_2^2$ \( ( 1 + 34 T^{2} + p^{2} T^{4} )^{2} \)
53$C_2^2$ \( ( 1 - 26 T^{2} + p^{2} T^{4} )^{2} \)
59$C_2^2$ \( ( 1 + 58 T^{2} + p^{2} T^{4} )^{2} \)
61$C_2$ \( ( 1 - 14 T + p T^{2} )^{2}( 1 + 14 T + p T^{2} )^{2} \)
67$C_2$ \( ( 1 + 10 T + p T^{2} )^{4} \)
71$C_2^2$ \( ( 1 - 17 T^{2} + p^{2} T^{4} )^{2} \)
73$C_2^2$ \( ( 1 - 38 T^{2} + p^{2} T^{4} )^{2} \)
79$C_2$ \( ( 1 - 2 T + p T^{2} )^{4} \)
83$C_2^2$ \( ( 1 + 106 T^{2} + p^{2} T^{4} )^{2} \)
89$C_2^2$ \( ( 1 + 43 T^{2} + p^{2} T^{4} )^{2} \)
97$C_2$ \( ( 1 - 14 T + p T^{2} )^{2}( 1 + 14 T + p T^{2} )^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−9.041912193689631135262666294743, −9.034931360600023406680396821073, −8.836656498878657218387275522995, −8.771312455738900040001387011517, −8.269443455592867292505575837509, −7.70462974402408316207458772645, −7.58381904159027412402719836443, −7.23712989446454673679922460951, −6.83104194364633565449063006377, −6.78709535216385006703237089919, −6.46441248412202969167806553322, −6.22291742876089152161715416877, −5.82526903388341409099158626797, −5.72926167363061150772777409765, −4.96082542505710999669918964658, −4.95738757290599019952705139486, −4.57893158611956600816912200405, −4.09457980771662169030647271265, −3.78254987793684010819448330215, −3.58618483410507751004618352609, −2.92167662656357775997283223476, −2.76691047005576429478913120071, −2.54379477291757560647864572511, −1.46341883486654928671535527742, −0.37132668927245395839836744855, 0.37132668927245395839836744855, 1.46341883486654928671535527742, 2.54379477291757560647864572511, 2.76691047005576429478913120071, 2.92167662656357775997283223476, 3.58618483410507751004618352609, 3.78254987793684010819448330215, 4.09457980771662169030647271265, 4.57893158611956600816912200405, 4.95738757290599019952705139486, 4.96082542505710999669918964658, 5.72926167363061150772777409765, 5.82526903388341409099158626797, 6.22291742876089152161715416877, 6.46441248412202969167806553322, 6.78709535216385006703237089919, 6.83104194364633565449063006377, 7.23712989446454673679922460951, 7.58381904159027412402719836443, 7.70462974402408316207458772645, 8.269443455592867292505575837509, 8.771312455738900040001387011517, 8.836656498878657218387275522995, 9.034931360600023406680396821073, 9.041912193689631135262666294743

Graph of the $Z$-function along the critical line